An issue with length contraction

[Dale]

Is there a length contraction, or is it only a different observation?

Yes to both. Those are not mutually exclusive

 

[student34]

Yes to both. Those are not mutually exclusive

I do not understand how the former is true too. Wouldn't a contraction have to involve the shortening of distance between two events kind of like we would need for any to points on a 2d object? I do not see any changes made between any 2 events anywhere in the example.

 

[PeterDonis]

the shortening of two events

You can't "shorten" events. Events are points in spacetime. They don't even have a "length" to begin with.

"Length contraction" just means that you have spacelike intervals between different pairs of events, that correspond to the "length" of an object according to different observers (because the events in each pair are on the worldlines of the two ends of the object). The spacelike interval between the pair of events corresponding to the length of the object in the frame in which it is moving, is shorter than the spacelike interval between the pair of events corresponding to the length of the object in the frame in which it is at rest.

 

[student34]

You can't "shorten" events. Events are points in spacetime. They don't even have a "length" to begin with.

"Length contraction" just means that you have spacelike intervals between different pairs of events, that correspond to the "length" of an object according to different observers (because the events in each pair are on the worldlines of the two ends of the object). The spacelike interval between the pair of events corresponding to the length of the object in the frame in which it is moving, is shorter than the spacelike interval between the pair of events corresponding to the length of the object in the frame in which it is at rest.

I changed edited my post to what I meant to say.

 

[PeterDonis]

I changed edited my post to what I meant to say.

The edited post isn't any better. You can't "shorten the distance between two events". The distance between two points in spacetime is an invariant. It makes no sense to talk about making it "shorter". Nor does "length contraction" involve any such thing. It involves comparing two different distances (spacetime intervals), between two different pairs of events, just as I described.

 

[student34]

You can't "shorten" events. Events are points in spacetime. They don't even have a "length" to begin with.

"Length contraction" just means that you have spacelike intervals between different pairs of events, that correspond to the "length" of an object according to different observers (because the events in each pair are on the worldlines of the two ends of the object). The spacelike interval between the pair of events corresponding to the length of the object in the frame in which it is moving, is shorter than the spacelike interval between the pair of events corresponding to the length of the object in the frame in which it is at rest.

Yes, the interval is shorter. But that only seems to be a matter of perception.

For example, Bob is observing the distance of one side of a square to be 1 meter, and Alice is observing opposite corners of the square to be 2^(1/2). We wouldn't say that the square contracted. Nothing is changing or contracting.

Furthermore, we wouldn't say that the square is shorter for Bob than it is for Alice. That is trivially incorrect.

 

[PeterDonis]

the interval is shorter. But that only seems to be a matter of perception.

I have no idea what you mean. Spacetime intervals are invariants. All observers agree on them.

For example, Bob is observing the distance of one side of a square to be 1 meter, and Alice is observing opposite corners of the square to be 2^(1/2). We wouldn't say that the square contracted. Nothing is changing or contracting.

That is true. But we also would not say that Alice was measuring a "side" of the square to begin with. We would say she was measuring a "diagonal". And everybody recognizes that the side and the diagonal of a square are different line segments with different lengths.

In the spacetime case, however, each observer, Alice and Bob, considers the interval they are measuring to be the "length" of the object. And since the "length" measured by one is shorter than the "length" measured by the other, the term "length contraction" is used. It is probably not the best term to describe the physics, but it's the term that's in all the literature so we're stuck with it.

Furthermore, we wouldn't say that the square is shorter for Bob than it is for Alice. That is trivially incorrect.

Of course, because we all recognize that the side of a square is shorter than its diagonal. Our intuitions on this are fine.

Our intuitions also, however, tell us that the "length" of an object should not depend on whether it is moving relative to us or not. Yet relativity tells us that it does. That is why "length contraction" is counterintuitive: because our intuitions are not fine in this case.

We could "fix" this "problem" by requiring another word to be used to describe measuring what we now call the "length" of an object if the object is moving relative to us. But changing the words we use wouldn't change the physics at all.

 

[Dale]

Is there a length contraction

A measurement of the length of an object involves measuring the distance between two ends of the object at the same time. Since two people in relative motion disagree on “at the same time” they disagree on the length of a given object. This is length contraction

Bob is observing the distance of one side of a square to be 1 meter, and Alice is observing opposite corners of the square to be 2^(1/2). We wouldn't say that the square contracted. Nothing is changing or contracting.

And yet it is a fact that there are holes that the square can fit through straight on that it cannot fit through diagonally.

Regarding the terminology. There is nothing we can do about that.

 

[student34]

Our intuitions also, however, tell us that the "length" of an object should not depend on whether it is moving relative to us or not. Yet relativity tells us that it does. That is why "length contraction" is counterintuitive: because our intuitions are not fine in this case.

Keeping in mind the diagonal/square analogy, and with all due respect, your statement about an object's length dependence seems to be deceptive and grows counterintuition unnecessarily. Maybe we should use "perceived length" or something like that?

 

[Dale]

Maybe we should use "perceived length" or something like that?

The terminology is not going to change, so that is not worth the angst. However, do you believe that the longer length across the diagonal of a square vs the side of a square is a matter of perception? Equivalently, if you are trying to fit a sofa through a doorway, is it a matter of perception if it doesn’t fit diagonally?

 

[PeterDonis]

your statement about an object's length dependence seems to be deceptive

Only if you refuse to actually look at the physics (i.e., the mathematical description, which is exact and unambiguous) and expect the ordinary language description to be enough. But it never is. Not for any area of physics. Ordinary language is never enough; you always need to look at the mathematical description if you want an exact, unambiguous description of the physics.

Maybe we should use "perceived length" or something like that?

Good luck persuading everyone who writes scientific papers on relativity. Not to mention having to go back and revise all of the past publications.

Also, you are ignoring the reason why the word "length" is used in ordinary language for both measurements: because, according to our intuitions, both measurements are measurements of the length of the object. Our intuitions are of course wrong in believing that the length measured by both measurements should be the same, since they aren't, but you can't fix wrong intuitions by finding better ordinary language descriptions. You fix them by learning the math.

 

[student34]

The terminology is not going to change, so that is not worth the angst. However, do you believe that the longer length across the diagonal of a square vs the side of a square is a matter of perception? Equivalently, if you are trying to fit a sofa through a doorway, is it a matter of perception if it doesn’t fit diagonally?

You are just changing the orientation. The couch itself does not contract.

 

[student34]

Only if you refuse to actually look at the physics (i.e., the mathematical description, which is exact and unambiguous) and expect the ordinary language description to be enough. But it never is. Not for any area of physics. Ordinary language is never enough; you always need to look at the mathematical description if you want an exact, unambiguous description of the physics.Good luck persuading everyone who writes scientific papers on relativity. Not to mention having to go back and revise all of the past publications.

Also, you are ignoring the reason why the word "length" is used in ordinary language for both measurements: because, according to our intuitions, both measurements are measurements of the length of the object. Our intuitions are of course wrong in believing that the length measured by both measurements should be the same, since they aren't, but you can't fix wrong intuitions by finding better ordinary language descriptions. You fix them by learning the math.

I want to understand all of this; I don't just want to know it.

 

[Dale]

You are just changing the orientation. The couch itself does not contract.

Nevertheless, the difference in length is not a matter of perception, is it?

 

[Dale]

Also, you are ignoring the reason why the word "length" is used in ordinary language for both measurements: because, according to our intuitions, both measurements are measurements of the length of the object.

@student34 this is a key point. See also my description of the measurement of length in post 43. It is what we normally think of as length.

 

[PeterDonis]

I want to understand all of this; I don't just want to know it.

Understand what?

If you mean understand the physics involved in what is called in vague ordinary language "length contraction", I think what has been posted in this thread already should be more than enough.

If you mean understand why people adopt vague ordinary language terms that don't always convey the actual physics very well, that's way off topic for this forum; it's a question of human psychology and sociology, not physics.

 

[Dale]

If you mean understand why people adopt vague ordinary language terms that don't always convey the actual physics very well, that's way off topic for this forum; it's a question of human psychology and sociology, not physics.

And history too.

 

[student34]

Nevertheless, the difference in length is not a matter of perception, is it?

No, but you/GR seem to be saying such a thing.

A more precise analogy is if I walk from in front of the couch to the side where it is say 1 meter wide, I shouldn't say that the couch contracted from x meters to 1 meter.

 

[student34]

Understand what?

If you mean understand the physics involved in what is called in vague ordinary language "length contraction", I think what has been posted in this thread already should be more than enough.

If you mean understand why people adopt vague ordinary language terms that don't always convey the actual physics very well, that's way off topic for this forum; it's a question of human psychology and sociology, not physics.

Ok, I agree, the thread has run its course as far as I am concerned. I just wanted to know if I was misinterpreting something about length contraction. Now I don't think I am.

 

[student34]

And history too.

Probably philosophy and linguistics too.

 

[PeterDonis]

you/GR seem to be saying such a thing.

No, we aren't. See below.

A more precise analogy is if I walk from in front of the couch to the side where it is say 1 meter wide, I shouldn't say that the couch contracted from x meters to 1 meter.

You wouldn't say that, but that's because you are talking about the couch's size along two different spatial directions, and we have different words for the couch's size along different spatial directions. You would say that the couch is, say, x meters long by 1 meter wide. Or x meters along a diagonal and 1 meter wide.

In the case of length contraction, we don't have different ordinary language words for the two different spacelike intervals. We use the same ordinary language word, "length", for both of them, because our ordinary language does not recognize time as a separate dimension, the way we recognize the two spatial dimensions of the couch or the square as separate dimensions. Our ordinary language only sees that in both cases we are measuring "length" along a single spatial direction, the ##x## direction. But that's just a limitation of our vague ordinary language.

Once we look beneath the vague ordinary language at the actual math and physics (geometry), we see the same thing in both cases: we have two different line segments, between two different pairs of points, that have different lengths. There is no "matter of perception": it's just straightforward geometry in both cases.

 

[PeterDonis]

I just wanted to know if I was misinterpreting something about length contraction. Now I don't think I am.

I'm not so sure. See my post #56 just now.

 

[Dale]

No, but you/GR seem to be saying such a thing.

A more precise analogy is if I walk from in front of the couch to the side where it is say 1 meter wide, I shouldn't say that the couch contracted from x meters to 1 meter.

The terminology is what it is. Contraction may not be the best word, but it is the word used to describe the measurements I described above. Those measurements are not just perception. They describe the measured geometry of physical objects.

If we could easily change bad terminology then we wouldn’t still have relativistic mass.

 

[renormalize]

So perhaps it would useful to look at cases where there are real physical implications, such as with muon decay observations...?

student34 should carefully ponder this physical example of length contraction (and time dilation):

Cosmic rays are known to strike the upper atmosphere at ##\sim15000\mathrm{m}## altitude, producing downward-directed muons traveling at nearly the speed-of-light ##c##. Since muons at rest have a mean lifetime of ##\overline{\tau}=2.2\mathrm{\mu s}##, their maximum travel distance in that time is only ##c\overline{\tau}=660\mathrm{m}##. This is much less than the distance to the ground, yet, due to relativity, these muons are readily detected at the Earth's surface.

In the muon's rest frame, the Earth is seen to approach at nearly ##c## with the depth of its atmosphere foreshortened (length-contracted) by a factor of ##1/\gamma##, where ##\gamma\approx21##. This is thin enough that the muon easily reaches the ground before decaying, in agreement with experiment. (See the left illustration below, where the depth of the atmosphere is symbolized by the height of the mountain.)

Alternatively, at rest on Earth we observe the "internal clock" of the muon to be slowed-down by time-dilation, increasing its apparent lifetime ##\overline{\tau}## by the same ##\gamma##-factor of ##\sim21##. This gives the muon ample time to reach the surface, again just as experimentally observed. (See the right illustration below.)

I leave it to the OP and the philosophers to debate whether this contraction of atmospheric-depth or dilation of half-life (depending on the reference frame) is "actual" or "apparent", but operationally both effects certainly seem "real" to me.

(Martin Bauer on Twitter)

 

[Ibix]

Maybe we should use "perceived length" or something like that?

Just to illustrate how silly worrying about terminology is, measure the length of something. Now measure it again. Now remember that you advanced in time between those two measurements, so you did not measure the same thing, but rather the intervals between two different pairs of events. So why are you calling them "the" length? They're different measurements even if the answers are the same, so shouldn't you also be agitating not to use the word length at all? We should always talk about the cross section of a worldtube measured at 08:49:23 on October 12th 2022 and the one measured three seconds later, and never the length.

This is why in physical sciences we reason in maths. "The" length is imprecise, let alone length contraction, and this will always be the case whatever term you try to introduce.

 

[Lluis Olle]

The terminology is what it is. Contraction may not be the best word, but it is the word used to describe the measurements I described above. Those measurements are not just perception. They describe the measured geometry of physical objects.

If we could easily change bad terminology then we wouldn’t still have relativistic mass.

I would like to point that a strain gauge inside the moving object would not record anything special, except for the acceleration and deacceleration phases perhaps (and some tiny geometric relativistic tension if is constantly accelerating), and that a beautiful object, designed by Leonardo da Vinci golden ratio proportions, would be apparently deformed in the direction of movement, so the "measures" of physical objects seem just perceptions of a geometry that changes its rules by the relative speed of observers. Is like looking through a glass. Nevertheless, time is other kind of stuff.

 

[Ibix]

a beautiful object, designed by Leonardo da Vinci with aural proportions,

I suspect a translation failure here. By "aural proportions" I guess you mean ##(1\pm\sqrt 5)/2##? That's called the Golden Ratio in English; aural means "related to hearing".

the "measures" of physical objects seem pretty must just perceptions of a geometry that changes its rules by the relative speed of observers.

Not really. You are just measuring different parts of a 4d entity when you use different frames. It's quite closely analogous to a unit square, which is 1 wide. Rotate yourself 45° and the square is now a diamond which is ##\sqrt 2## wide. Neither the rules of geometry nor the square have changed, we just changed our mind about what we were calling "width". The same happens when we change frame in relativity - we change our mind about what space is, and hence about the intersection of the object's worldtube and space.

General comment, not just to Lluis Olle: note that there's potential for confusion over the word "object". Is it the 4d worldtube, or the 3d slice of the worldtube that intersects our "now"? Make sure you know which one you mean and make sure you consider the possibility that someone you are talking to means the other one.

 

[Lluis Olle]

I suspect a translation failure here. By "aural proportions" I guess you mean ##(1\pm\sqrt 5)/2##? That's called the Golden Ratio in English; aural means "related to hearing".

Sure, I mean the golden ratio, which is "la proporción aurea" for me. Is a false friend word :) Here you have a relativistic effect in action.

Rotate yourself 45° and the square is now a diamond which is 2 wide. Neither the rules of geometry nor the square have changed

It continues to be a square, all its four sides are equal (I don't say the same length, I mean that can't be distinguished and are interchangeable).

 

[Ibix]

It continues to be a square, all its four sides are equal (I don't say the same length, I mean that can't be distinguished and are interchangeable).

Exactly. But the width is different because it's not purely a property of the square: width also depends on the orientation of the square/diamond with respect to you. Similarly, length contraction arises because you measure a different cross section of the worldtube and still call it length.

 

[Lluis Olle]

Exactly. But the width is different because

Sorry, what is "the width"? ... a square is "a plane figure with four equal straight sides and four right angles".

And by cross section you mean this? Because being this a Physics forum, some words like "fragile", and others have an agreed meaning in the community.

 

[Ibix]

Sorry, what is "the width"? ... a square is "a plane figure with four equal straight sides and four right angles".

It depends which way you're looking at it, which is my point. If we've drawn the figure on this screen and you're holding your screen vertically, I'd say it's the horizontal extent of it in the plane of your screen. Rotate your screen in its own plane to vary the width.

And by cross section you mean this?

I mean that "the object, now" is the intersection between a worldtube and a flat spacelike plane, in particular the one you are calling "now" in your frame. That's a cross-section in the same sense that a slice through a sausage gives you a circular or elliptical cross section of the cylindrical sausage.

 

[jbriggs444]

Sorry, what is "the width"? ... a square is "a plane figure with four equal straight sides and four right angles".

I think that a "square" it not as helpful an analogy as could have been achieved. A "two-edged measuring tape" would be better so that we do not have any distracting ends. Just the two sides.

How do we measure the width of a tape? By placing a ruler on it crosswise.
But what if the ruler is placed at an angle? We obtain a different measurement.

So we are careful. We place the ruler on the tape so that it is at the "1 centimeter" marker on both edges of the tape.

One would be tempted to use a try-square to measure this Euclidean width but relativity does not provide us with the equivalent of a "right angle" between space and time, so we are forced to this alternate mechanism in order to keep the analogy apt.

Then we worry about the careful details of the alignment in markings on the two edges of the tape. This is analogous to the relativity of simultaneity.

 

[student34]

student34 should carefully ponder this physical example of length contraction (and time dilation):

Cosmic rays are known to strike the upper atmosphere at ##\sim15000\mathrm{m}## altitude, producing downward-directed muons traveling at nearly the speed-of-light ##c##. Since muons at rest have a mean lifetime of ##\overline{\tau}=2.2\mathrm{\mu s}##, their maximum travel distance in that time is only ##c\overline{\tau}=660\mathrm{m}##. This is much less than the distance to the ground, yet, due to relativity, these muons are readily detected at the Earth's surface.

In the muon's rest frame, the Earth is seen to approach at nearly ##c## with the depth of its atmosphere foreshortened (length-contracted) by a factor of ##1/\gamma##, where ##\gamma\approx21##. This is thin enough that the muon easily reaches the ground before decaying, in agreement with experiment. (See the left illustration below, where the depth of the atmosphere is symbolized by the height of the mountain.)

Alternatively, at rest on Earth we observe the "internal clock" of the muon to be slowed-down by time-dilation, increasing its apparent lifetime ##\overline{\tau}## by the same ##\gamma##-factor of ##\sim21##. This gives the muon ample time to reach the surface, again just as experimentally observed. (See the right illustration below.)

I leave it to the OP and the philosophers to debate whether this contraction of atmospheric-depth or dilation of half-life (depending on the reference frame) is "actual" or "apparent", but operationally both effects certainly seem "real" to me.

View attachment 315451
(Martin Bauer on Twitter)

It is still sort of an illusion. The muon just takes a shorter route through spacetime to get to the ground.

If we were to make a time over distance graph and display the muon trip horizontally, we see that the event where the muon takes off from and the event where the muon impacts the Earth are along two parallel lines extending only through time (as seen by an observer at rest). Then the world line of the muon just takes a shorter spacetime interval to get to the Earth.

The muon is the green worldline. And some object that takes off where the muon takes off from, but at a slower speed, is the blue line.

 

[Ibix]

It is still sort of an illusion. The muon just takes a shorter route through spacetime to get to the ground.

Why is the green line more illusory than the blue one?

 

[PeterDonis]

The muon just takes a shorter route through spacetime to get to the ground.

Shorter compared to what?

ome object that takes off where the muon takes off from, but at a slower speed

There is no such object in the scenario. What's the point of this?